ON A NOETHERIAN SCHEME
LARS WINTHER CHRISTENSEN, SERGIO ESTRADA, AND PEDER THOMPSON
Abstract. For a semi-separated noetherian scheme, we show that the cate- gory of cotorsion Gorenstein flat quasi-coherent sheaves is Frobenius and a nat- ural non-affine analogue of the category of Gorenstein projective modules over a noetherian ring. We show that this coheres perfectly with the work of Mur- fet and Salarian that identifies the pure derived category of F-totally acyclic complexes of flat quasi-coherent sheaves as the natural non-affine analogue of the homotopy category of totally acyclic complexes of projective modules.
Introduction
A classic result due to Buchweitz [2] says that the singularity category of a Goren- stein local ring A is equivalent to the homotopy category Ktac(prj(A)) of totally acyclic complexes of finitely generated projectiveA-modules. The latter category is also equivalent to the stable category of finitely generated maximal Cohen-Macaulay A-modules or, in a different terminology, to the stable categoryStGprj(A) of finitely generated Gorenstein projective A-modules. This second equivalence extends be- yond the realm of Gorenstein local rings and finitely generated modules: For ev- ery ring A, the category Ktac(Prj(A)) of totally acyclic complexes of projective A-modules is equivalent to the stable categoryStGPrj(A) of Gorenstein projective A-modules. We obtain this folklore result as a special case of [5, Corollary 3.9].
What is the analogue in the non-affine setting?
Murfet and Salarian [23] offer a non-affine analogue of the categoryKtac(Prj(A)) over a semi-separated noetherian schemeX in the form of the Verdier quotient,
DF-tac(Flat(X)) = KF-tac(Flat(X)) Kpac(Flat(X)) ,
of the homotopy category of F-totally acyclic complexes of flat quasi-coherent sheaves onX by its subcategory of pure-acyclic complexes. Indeed, for a commu- tative noetherian ringAof finite Krull dimension andX = Spec(A), the categories Ktac(Prj(A)) and DF-tac(Flat(X)) are equivalent by [23, Lemma 4.22]. What re- mains is to identify an analogue of the categoryStGPrj(A) in the non-affine setting, and that is the goal of this paper.
Date: 22 May 2020.
2020Mathematics Subject Classification. 14F08; 18G35.
Key words and phrases. Cotorsion sheaf, Gorenstein flat sheaf, noetherian scheme, stable category, totally acyclic complex.
L.W.C. was partly supported by Simons Foundation collaboration grant 428308. S.E. was partly supported by grants PRX18/00057, MTM2016-77445-P, and 19880/GERM/15 by the Fun- daci´on S´eneca-Agencia de Ciencia y Tecnolog´ıa de la Regi´on de Murcia and FEDER funds.
The stable category of Gorenstein projective modules is a standard construction that applies to any Frobenius category. The category of Gorenstein flat modules is rarely Frobenius; it is essentially only Frobenius if it coincides with the category of Gorenstein projective modules, see [5, Theorem 4.5]. The cotorsion Gorenstein flat modules, however, do form a Frobenius category, and a special case of [5, Corol- lary 5.9] says that for a commutative noetherian ringA of finite Krull dimension, the category StGPrj(A) is equivalent to the stable categoryStGFC(A) of cotorsion Gorenstein flat modules. This identifies a candidate category and, indeed, the goal stated above is obtained (in 4.6) with
Theorem A. Let X be a semi-separated noetherian scheme. The stable category StGFC(X)of cotorsion Gorenstein flat sheaves is equivalent toDF-tac(Flat(X)).
In the statement of this theorem, and everywhere else in this paper, a sheaf means a quasi-coherent sheaf. A sheaf onX is called cotorsion if it is right Ext-orthogonal to flat sheaves onX.
A crucial step towards the equivalence in Theorem A is to prove (in 4.2 and 4.3) that the sheaves that are both cotorsion and Gorenstein flat are precisely the sheaves that arise as cycles inF-totally acyclic complexes of flat-cotorsion sheaves.
This is exactly what happens in the affine case, and it transpires that the main take-away from [5] also applies in the non-affine setting: One should work with sheaves that are both cotorsion and Gorenstein flat rather than all Gorenstein flat sheaves! One manifestation is a result (4.7) that sharpens [23, Theorem 4.27]:
Theorem B. A semi-separated noetherian scheme X is Gorenstein if and only if every acyclic complex of flat-cotorsion sheaves onX is F-totally acyclic.
A second manifestation—actually the result behind Theorem A—is that the category DF-tac(Flat(X)) considered by Murfet and Salarian is equivalent to the homotopy categoryKF-tac(Flat(X)∩Cot(X)) of F-totally acyclic complexes of flat cotorsion sheaves (see 4.5). That is, passing from the affine to the non-affine setting, one can replace the homotopy categoryKtac(Prj(A)) by another homotopy category.
Up to equivalence, the category KF-tac(Flat(X)∩Cot(X)) arises in a related, yet different, context. The category of Gorenstein flat sheaves on a semi-separated noetherian schemeX is part of a complete hereditary cotorsion pair (see 2.2), and one that is comparable to the cotorsion pair of flat sheaves and cotorsion sheaves onX. Through work of Hovey [21] and Gillespie [16], these cotorsion pairs induce a model structure on the category of sheaves on X. We prove (see 4.4) that the associated homotopy category is equivalent toKF-tac(Flat(X)∩Cot(X)).
1. Gorenstein flat sheaves
In this paper, the symbol X denotes a scheme with structure sheaf OX. By a sheaf onX we shall always mean a quasi-coherent sheaf, andQcoh(X) denotes the category of (quasi-coherent) sheaves onX. We frequently add the assumption that X issemi-separated, by which we mean thatX has an open affine covering U such thatU∩V is affine for allU, V ∈ U; such a covering is referred to assemi-separating.
We use standard cohomological notation for cochain complexes.
In this first section we show that over a semi-separated noetherian scheme, one can equivalently define Gorenstein flatness of sheaves globally, locally, or stalkwise.
Let A be a commutative ring. An acyclic complex F of flat A-modules is called
F-totally acyclic if the complexI⊗AF is acyclic for every injective A-moduleI.
AnA-module M is Gorenstein flat if there exists an F-totally acyclic complexF withM = Z0(F). Denote byGFlat(A) the category of Gorenstein flatA-modules.
Remark 1.1. For an acyclic complexF of flat sheaves on X there is a global, a local, and a stalkwise notion of F-total acyclicity:
• For every injective sheafI onX the complexI ⊗F is acyclic.
• For every open affineU ⊆XtheOX(U)-complexF(U) isF-totally acyclic.
• For everyx∈X theOX,x-complexFx isF-totally acyclic.
It is proved in [23, Lemmas 4.4 and 4.5] that all three notions agree if the scheme X is semi-separated noetherian. Christensen, Estrada, and Iacob [4, Corollary 2.8]
show that the local notion is Zariski-local, and by [4, Proposition 2.10] the local and global notions agree ifXis semi-separated and quasi-compact (which is weaker than noetherian).
Definition 1.2. Assume thatXis semi-separated noetherian. An acyclic complex F of flat sheaves onX is calledF-totally acyclic if it satisfies the equivalent con- ditions in Remark 1.1. A sheafM onX is called Gorenstein flatif there exists an F-totally acyclic complex F of flat sheaves on X with M = Z0(F). Denote by GFlat(X) the category of Gorenstein flat sheaves onX.
Over any scheme, Gorenstein flatness can also be defined locally or stalkwise, and we proceed to show that these notions agree with Gorenstein flatness as defined above if the scheme is semi-separated noetherian.
Definition 1.3. A sheafM onX is calledlocally Gorenstein flat if for every open affine subsetU ⊆X theOX(U)-moduleM(U) is Gorenstein flat, andM is called stalkwise Gorenstein flat ifMx is a Gorenstein flatOX,x-module for everyx∈X. Like localF-total acyclicity, local Gorenstein flatness is a Zariski-local property, at least under mild assumptions on the scheme. As shown in [4], this follows from the next proposition.
Proposition 1.4. Let ϕ:A→B be a flat homomorphism of commutative rings.
(a) If M is a Gorenstein flat A-module, then B⊗AM is a Gorenstein flat B-module.
(b) Assume that A is coherent and ϕ is faithfully flat. An A-module M is Gorenstein flat if theB-module B⊗AM is Gorenstein flat.
Proof. (a) LetFbe anF-totally acyclic complex of flatA-modules withM = Z0(F).
By [4, Proposition 2.7(1)] the B-complexB⊗AF is an F-totally acyclic complex of flatB-modules, soB⊗AM = Z0(B⊗AF) is a Gorenstein flatB-module.
(b) It follows from work of ˇSaroch and ˇS ’tov´ıˇcek [28, Corollary 4.12] that the categoryGFlat(A) is closed under extensions, so the assertion is immediate from a result of Christensen, K¨oksal, and Liang [6, Theorem 1.1].
Corollary 1.5. Assume that X is locally coherent. A sheaf M on X is locally Gorenstein flat if there exists an open affine coveringU ofX such that theOX(U)- module M(U)is Gorenstein flat for every U ∈ U.
Proof. Proposition 1.4 shows that Gorenstein flatness is an ascent–descent property for modules over commutative coherent rings. Now invoke [4, Lemma 2.4].
Theorem 1.6. Assume that X is semi-separated noetherian. For a sheaf M on X the following conditions are equivalent.
(i) M is Gorenstein flat.
(ii) M is locally Gorenstein flat.
(iii) M is stalkwise Gorenstein flat.
Proof. The implication (i)⇒(ii) is trivial by the definition of Gorenstein flatness;
see Remark 1.1.
(ii)⇒ (iii): Letx∈ X and choose an open affine subsetU ⊆X with x∈U. Localization is exact and commutes with tensor products, so it preserves Gorenstein flatness of modules, whence the moduleMx∼=M(U)x is Gorenstein flat over the local ringOX,x∼=OX(U)x.
(iii) ⇒ (i): This argument is inspired by Yang and Liu [29, Lemmas 3.8 and 3.9]. Let U = {U0, . . . , Un} be a semi-separating open affine covering ofX. For everyx∈X there is a short exact sequence ofOX,x-modules,
(1) 0−→Mx−→Fx−→Tx−→0,
where Fx is flat and Tx is Gorenstein flat. For x ∈ X and U ∈ U consider the canonical maps
ix: Spec(OX,x)−→X and iU: Spec(OX(U))−→X; for x ∈ U the map ix factors through iU. The map M → Q
x∈X(ix)∗(gMx) is a monomorphism locally at everyy∈X, as one has
Y
x∈X
(ix)∗(gMx)∼= (iy)∗(gMy)⊕ Y
x∈X\{y}
(ix)∗(gMx),
so it is a monomorphism inQcoh(X). Now (1) yields a monomorphism M −→ Y
x∈X
(ix)∗(fFx),
so withF0=Q
x∈X(ix)∗(fFx) there is an exact sequence inQcoh(X)
(2) 0−→M −→F0−→K1−→0.
The first goal is to show thatF0is flat. For everyx∈Xchoose only one element Uk in U with x∈Uk, and fork = 0, . . . , nlet Ik ⊆Uk denote the corresponding subset such thatX is the disjoint unionSn
k=0Ik. Now one has F0= Y
x∈X
(ix)∗(fFx) =
n
M
k=0
Y
x∈Ik
(iUk)∗(fFx)∼=
(†)∼=
n
M
k=0
(iUk)∗(Y
x∈Ik
Ffx)∼=
n
M
k=0
(iUk)∗(Y^
x∈Ik
Fx),
where the isomorphism (†) holds as (iUk)∗, being a right adjoint functor, preserves direct products. SinceFx is a flat OX(Uk)-module, and OX(Uk) is noetherian, it follows thatQ
x∈IkFx is a flatOX(Uk)-module. HenceF0 is a flat sheaf.
The second goal is to show that K1 is Gorenstein flat locally at every point y∈X. Consider the commutative diagram ofOX,y-modules
0 //My //Fy0 //
π
Ky1 //
$
0
0 //My //Fy //Ty //0
whereπis the canonical projection with kernelL; this is a flat module as it is the kernel of an epimorphism between flatOX,x-modules. By the Snake Lemma $ is surjective with kernelL, so Ky1 is Gorenstein flat; see e.g. [28, Corollary 4.12].
LetI be an injective sheaf onX; we argue that (2) remains exact after tensoring withI by showing that TorQcoh(X)1 (I,K1) = 0 holds. Forx∈X letJ(x) be the sheaf on Spec(OX,x) associated to the injective hull of the residue field of the local ringOX,x. One has
I ∼=M
x∈X
(ix)∗J(x)(Λx)
for some index sets Λx; see Hartshorne [19, Proposition II.7.17]. Therefore, it suffices to verify that TorQcoh(X)1 ((ix)∗J(x),K1) = 0 holds for every x∈X. This can be verified locally, and every localization TorQcoh(X)1 ((ix)∗J(x),K1)x0 is 0 or isomorphic to TorO1X,x(J(x),Kx1), and the latter is also 0 as Kx1 is a Gorenstein flatOX,x-module.
Repeating this process, one gets an exact sequence of sheaves (3) 0−→M −→F0−→F1−→F2−→ · · ·
which remains exact after tensoring with any injective sheaf on X. Since X, in particular, is semi-separated quasi-compact, every sheaf is a homomorphic image of a flat sheaf; see for example Efimov and Positselski [7, Lemma A.1]. Therefore, there is an exact sequence
(4) 0−→K−1−→F−1−→M −→0
withF−1 a flat sheaf. The class of Gorenstein flat modules is closed under kernels of epimorphisms, see e.g. [28, Corollary 4.12], so Kx−1 is a Gorenstein flat OX,x- module for everyx∈X. By the same argument as above the sequence (4) remains exact after tensoring with any injective sheaf on X. Repeating this process, one obtains an exact sequence
(5) · · · −→F−3−→F−2−→F−1−→M −→0
that remains exact after tensoring with any injective sheaf onX. Splicing together (3) and (5) one gets per Definition 1.2 anF-totally acyclic complex of flat sheaves, F =· · · →F−1→F0→F1→ · · ·. Thus,M = Z0(F) is Gorenstein flat.
Henceforth we work mainly over semi-separated noetherian schemes. In that set- ting we consistently refer to the sheaves described in Theorem 1.6 by their shortest name: Gorenstein flat; some proofs, though, rely crucially on their local properties.
2. The Gorenstein flat model structure on Qcoh(X)
Let G be a Grothendieck category, that is, an abelian category that has colimits, exact direct limits (filtered colimits), and a generator. A class C of objects in G is called resolving if it contains all projective objects and is closed under exten- sions and kernels of epimorphisms. To a classC of objects in G one associates the orthogonal classes
C⊥ = {G∈ G |Ext1G(C, G) = 0 for allC∈ C } and
⊥C = {G∈ G |Ext1G(G, C) = 0 for allC∈ C }.
LetS ⊆ Cbe a set. The pair (C,C⊥) is said to begenerated by the setSif an object Gbelongs toC⊥ if and only if Ext1G(C, G) = 0 holds for allC∈ S. A pair (F,C) of classes in G withF⊥ =C and ⊥C=F is called a cotorsion pair. The intersection F ∩ C is called thecore of the cotorsion pair.
A cotorsion pair (F,C) in G is calledhereditary if for allF ∈ F andC∈ C one has ExtiG(F, C) = 0 for alli≥1. Notice that the class F in this case is resolving.
A cotorsion pair (F,C) in G is called complete provided that for every G∈ G there are short exact sequences 0→C→F →G→0 and 0→G→C0→F0→0 withF, F0 ∈ F andC, C0 ∈ C.
Abelian model category structures from cotorsion pairs. Gillespie [16]
shows how to construct a hereditary abelian model structure onG from two com- parable cotorsion pairs. Namely, if (Q,R) and (e Q,e R) are complete hereditary cotorsion pairs in G with R ⊆ R,e Q ⊆ Q, ande Q ∩Re =Q ∩ R, then there existse a unique thick (i.e. full, closed under direct summands, and having the two-out-of- three property) subcategory W of G such thatQe =Q ∩ W and Re =R ∩ W. In other words (Q,W,R) is a so-calledHovey triple, and from work of Hovey [21] it is known that there is a unique abelian model structure onG in whichQ, R, and Ware the classes of cofibrant, fibrant, and trivial objects, respectively; refer to [21]
for this standard terminology. We are now going to apply this machine to cotorsion pairs withQe andQthe categories of flat and Gorenstein flat sheaves on X.
Remark 2.1. If X is semi-separated quasi-compact, then Qcoh(X) is a locally finitely presentable Grothendieck category. This was proved already in EGA [18, I.6.9.12], though not using that terminology. Being a Grothendieck category,Qcoh(X) has a generator and hence, by [7, Lemma A.1], a flat generator. Sl´avik and ˇS ’tov´ıˇcek [26] have recently proved that if X is quasi-separated and quasi-compact, then Qcoh(X) has a flat generator if and only ifX is semi-separated.
Theorem 2.2. Assume that X is semi-separated noetherian. The pair (GFlat(X),GFlat(X)⊥)
is a complete hereditary cotorsion pair.
Proof. For an open affine subset U ⊆ X we write GFlat(U) for the category GFlat(OX(U)) of Gorenstein flat OX(U)-modules. For every open affine subset U ⊆X the pair (GFlat(U),GFlat(U)⊥) is a complete hereditary cotorsion pair; see Enochs, Jenda, and L´opez-Ramos [10, Theorems 2.11 and 2.12]. The proof of [10, Theorem 2.11] shows that the pair is generated by a set SU; see also the more precise statement in [28, Corollary 4.12].
A result of Estrada, Guil Asensio, Prest, and Trlifaj [11, Corollary 3.15] now shows that (GFlat(X),GFlat(X)⊥) is a complete cotorsion pair. Indeed, the flat generator of Qcoh(X) belongs to GFlat(X). As the quiver in [11, Notation 3.12]
one takes the quiver with vertices all open affine subsets ofX, and the class L in [11, Corollary 3.15] is in this case
L={L ∈Qcoh(X)|L(U)∈ SU for every open affine subsetU ⊆X}. Moreover since (GFlat(U),GFlat(U)⊥) is hereditary, the classGFlat(U) is resolving for every open affine subset U ⊆ X. It follows that GFlat(X) is also resolving, whence (GFlat(X),GFlat(X)⊥) is hereditary asGFlat(X) contains a generator; see
Saor´ın and ˇS ’tov´ıˇcek [25, Lemma 4.25].
LetX be a semi-separated noetherian scheme. ByFlat(X) we denote the cate- gory of flat sheaves onX. The proof of the next result is modeled on an argument due to Estrada, Iacob, and P´erez [12, Proposition 4.1].
Lemma 2.3. Assume thatX is semi-separated noetherian. In Qcoh(X)one has GFlat(X)∩GFlat(X)⊥=Flat(X)∩Flat(X)⊥.
Proof. “⊆”: Let M ∈GFlat(X)∩GFlat(X)⊥. The inclusionFlat(X)⊆GFlat(X) yieldsGFlat(X)⊥⊆Flat(X)⊥, so it remains to show thatM is flat. SinceM is in GFlat(X) there is an exact sequence inQcoh(X),
0−→M −→F −→N −→0,
with F a flat sheaf and N a Gorenstein flat sheaf on X. Since M belongs to GFlat(X)⊥ the sequence splits, whenceM is flat.
“⊇”: LetM ∈Flat(X)∩Flat(X)⊥. As the inclusionFlat(X)⊆GFlat(X) holds, it remains to show that M is in GFlat(X)⊥. Since (GFlat(X),GFlat(X)⊥) is a complete cotorsion pair inQcoh(X), see Theorem 2.2, there is an exact sequence inQcoh(X),
(6) 0−→M −→G −→N −→0,
withG ∈GFlat(X)⊥ andN ∈GFlat(X). Moreover, sinceGFlat(X) is closed under extensions by Theorem 2.2, also G belongs to GFlat(X). Thus the sheaf G is in GFlat(X)∩GFlat(X)⊥, so by the containment already provedG is flat. SinceM is also flat, it follows that
flat dimOX(U)N (U)≤1
holds for every open affine subsetU ⊆X. ThusN(U) is a Gorenstein flatOX(U)- module of finite flat dimension and, therefore, flat; see [9, Corollary 10.3.4]. It follows thatN is a flat sheaf. Since M ∈Flat(X)⊥ by assumption, the sequence (6) splits. Therefore,M is a direct summand of G and thus in GFlat(X)⊥. We call sheaves in the subcategory Cot(X) =Flat(X)⊥ of Qcoh(X) cotorsion.
Sheaves in the intersectionFlat(X)∩Cot(X) are calledflat-cotorsion.
Remark 2.4. Assume that X is semi-separated quasi-compact. In this case the category Flat(X) contains a generator for Qcoh(X), so it follows from work of Enochs and Estrada [8, Corollary 4.2] that (Flat(X),Cot(X)) is a complete cotorsion pair, and since Flat(X) is resolving it follows from [25, Lemma 4.25] that the pair (Flat(X),Cot(X)) is hereditary. This fact can also be deduced from work of Gillespie [14, Proposition 6.4] and Hovey [21, Corollary 6.6].
The next theorem establishes what we call the Gorenstein flat model structure onQcoh(X); it may be regarded as a non-affine version of [17, Theorem 3.3].
Theorem 2.5. Assume thatXis semi-separated noetherian. There exists a unique abelian model structure onQcoh(X)withGFlat(X)the class of cofibrant objects and Cot(X)the class of fibrant objects. In this structureFlat(X)is the class of trivially cofibrant objects andGFlat(X)⊥ is the class of trivially fibrant objects.
Proof. It follows from Theorem 2.2 and Remark 2.4, that (GFlat(X),GFlat(X)⊥) and (Flat(X),Cot(X)) are complete hereditary cotorsion pairs. Every flat sheaf is Gorenstein flat, and by Lemma 2.3 the two pairs have the same core, so they satisfy the conditions in [16, Theorem 1.2]. Thus the pairs determine a Hovey triple, and by [21, Theorem 2.2] a unique abelian model category structure on Qcoh(X) with
fibrant and cofibrant objects as asserted.
Corollary 2.6. Assume that the scheme X is semi-separated noetherian. The categoryCot(X)∩GFlat(X)is Frobenius and the projective–injective objects are the flat-cotorsion sheaves. Its associated stable category is equivalent to the homotopy category of the Gorenstein flat model structure.
Proof. Applied to the Gorenstein flat model structure from the theorem, [15, Propo- sition 5.2(4)] shows thatCot(X)∩GFlat(X) is a Frobenius category with the stated projective–injective objects. The last assertion follows from [15, Corollary 5.4].
3. Acyclic complexes of cotorsion sheaves
We assume throughout this section that X is semi-separated quasi-compact. The category of cochain complexes of sheaves on X is denoted C(Qcoh(X)). The goal is to establish a result, Theorem 3.3 below, which in the affine case is proved by Bazzoni, Cort´es Izurdiaga, and Estrada [1, Theorem 1.3]. It says, in part, that every acyclic complex of cotorsion sheaves has cotorsion cycles. Our proof is inspired by arguments of Hosseini [20] and ˇS ’tov´ıˇcek [27].
LetCZac(Flat(X)) denote the full subcategory ofC(Qcoh(X)) whose objects are the acyclic complexes F of flat sheaves with Zn(F) ∈ Flat(X) for every n ∈ Z; similarly, letCZac(Cot(X)) denote the full subcategory whose objects are the acyclic complexesC of cotorsion sheaves with Zn(C)∈Cot(X) for everyn∈Z. Further, Csemi(Cot(X)) denotes the category of complexes C of cotorsion sheaves with the property that the total Hom complex Hom(F,C) of abelian groups is acyclic for every complex F ∈CZac(Flat(X)). In the literature such complexes are referred to as dg- or semi-cotorsion complexes; it is part of Theorem 3.3 that every complex of cotorsion sheaves onX has this property.
Remark 3.1. The pair (CZac(Flat(X)),Csemi(Cot(X))) is by [14, Theorem 6.7] and [21, Theorem 2.2] a complete cotorsion pair inC(Qcoh(X)).
For complexesA andB of sheaves onX, let Hom(A,B) denote the standard total Hom complex of abelian groups. There is an isomorphism
(7) Ext1C(Qcoh(X)),dw(A,Σn−1B)∼= HnHom(A,B),
where Ext1C(Qcoh(X)),dw(A,Σn−1B) is the subgroup of Ext1C(Qcoh(X))(A,Σn−1B) consisting of degreewise split short exact sequences; see e.g. [13, Lemma 2.1]. For
a complexF of flat sheaves and a complexC of cotorsion sheaves, every extension 0→C →X →F →0 is degreewise split, so (7) reads
(8) Ext1C(Qcoh(X))(F,Σn−1C)∼= HnHom(F,C).
Lemma 3.2. Let(Fλ)λ∈Λ be a direct system of complexes inCZac(Flat(X)). If each complex Fλ is contractible, then
Ext1C(Qcoh(X))(lim
λ∈Λ−→
Fλ,C)∼= H1Hom(lim
λ∈Λ−→
Fλ,C) = 0 holds for every complexC of cotorsion sheaves onX.
Proof. The categoryQcoh(X) is locally finitely presentable and, therefore, finitely accessible; see Remark 2.1. It follows that the results, and arguments, in [27] apply.
The argument in the proof of [27, Proposition 5.3] yields an exact sequence,
(9) 0−→K −→M
λ∈Λ
Fλ−→lim
λ∈Λ−→
Fλ−→0,
where K is filtered by finite direct sums of complexes Fλ. That is, there is an ordinal number β and a filtration (Kα | α < β), where K0 = 0, Kβ = K, and Kα+1/Kα∼=L
λ∈JαFλ forα < β andJα a finite set.
LetC be a complex of cotorsion sheaves onX. AsCZac(Flat(X)) is closed under direct limits, one has lim
−→Fλ∈CZac(Flat(X)). Thus, Ext1Qcoh(X)((lim
−→Fλ)i,Cj) = 0 holds for all i, j ∈ Z, whence there is an exact sequence of complexes of abelian groups:
(10) 0−→Hom(lim
λ∈Λ−→
Fλ,C)−→Hom(M
λ∈Λ
Fλ,C)−→Hom(K,C)−→0.
By (8) it now suffices to show that the left-hand complex in this sequence is acyclic.
The middle complex is acyclic because each complexFλ and, therefore, the direct sumLFλ is contractible. Thus it is enough to prove that Hom(K,C) is acyclic.
SinceFlat(X) is resolving, it follows from (9) thatK is a complex of flat sheaves.
AsC is a complex of cotorsion sheaves, (8) yields
HnHom(K,C)∼= Ext1C(Qcoh(X))(K,Σn−1C).
Hence, it suffices to show that Ext1C(Qcoh(X))(K,Σn−1C) = 0 holds for all n∈Z. Let (Kα|α≤λ) be the filtration of K described above. For everyn∈Zone has
Ext1C(Qcoh(X))(M
λ∈Jα
Fλ,Σn−1C) = 0,
so Eklof’s lemma [27, Proposition 2.10] yields Ext1C(Qcoh(X))(K,Σn−1C) = 0.
Theorem 3.3. Assume that X is semi-separated quasi-compact. Every complex of cotorsion sheaves on X belongs to Csemi(Cot(X)), and every acyclic complex of cotorsion sheaves belongs to CZac(Cot(X)).
Proof. As (CZac(Flat(X)),Csemi(Cot(X))) is a cotorsion pair, see Remark 3.1, the first assertion is that for every complex M of cotorsion sheaves and every F in CZac(Flat(X)) one has Ext1C(Qcoh(X))(F,M) = 0. Fix F ∈ CZac(Flat(X)) and a semi-separating open affine covering U ={U0, . . . , Ud} ofX. Consider the double complex of sheaves obtained by taking the ˇCech resolutions of each term in F;
see Murfet [22, Section 3.1]. The rows of the double complex form a sequence in C(Qcoh(X)):
(11) 0−→F −→C0(U,F)−→C1(U,F)−→ · · · −→Cd(U,F)−→0 with
Cp(U,F) = M
j0<···<jp
i∗(F(U^j0,...,jp)),
wherej0, . . . , jp belong to the set{0, . . . , d} andi: Uj0,...,jp−→X is the inclusion of the open affine subset Uj0,...,jp = Uj0 ∩. . .∩Ujp of X. For a tuple of indices j0<· · ·< jp, the complexF(Uj0,...,jp) is an acyclic complex of flatOX(Uj0,...,jp)- modules whose cycle modules are also flat. It follows that the complexF(Uj0,...,jp) is a direct limit
F(Uj0,...,jp) = lim
λ∈Λ−→
PλUj0,...,jp
of contractible complexes of projective, hence flat, OX(Uj0,...,jp)-modules; see for example Neeman [24, Theorem 8.6]. The functori∗preserves split exact sequences, so i∗(Pλ^Uj0,...,jp) is for every λ ∈ Λ a contractible complex of flat sheaves. The functor also preserves direct limits, so Cp(U,F) is a finite direct sum of direct limits of contractible complexes inCZac(Flat(X)), henceCp(U,F) is itself a direct limit of contractible complexes inCZac(Flat(X)). For every complexM of cotorsion sheaves and everyn∈Z, Lemma 3.2 now yields
HnHom(Cp(U,F),M)∼= H1Hom(Cp(U,F),Σn−1M) = 0 for 0≤p≤d . That is, the complex Hom(Cp(U,F),M) is acyclic for every 0≤p≤dand every complexM of cotorsion sheaves. Applying Hom(−,M) to the exact sequence
0−→Ld−1−→Cd−1(U,F)−→Cd(U,F)−→0, one gets an exact sequence of complexes of abelian groups
0−→Hom(Cd(U,F),M)−→Hom(Cd−1(U,F),M)−→Hom(Ld−1,M)−→0. The first two terms are acyclic, and hence so is Hom(Ld−1,M). Repeating this argumentd−1 more times, one concludes that Hom(F,M) is acyclic, whence one has Ext1C(Qcoh(X))(F,M) = 0 per (8).
The second assertion now follows from [14, Corollary 3.9] which applies as (Flat(X),Cot(X)) is a complete hereditary cotorsion pair and Flat(X) contains a
generator forQcoh(X); see Remark 2.1.
4. The stable category of Gorenstein flat-cotorsion sheaves In this last section, we give a description of the stable category associated to the cotorsion pair of Gorenstein flat sheaves described in Theorem 2.2. In particular, we prove Theorems A and B from the introduction.
Here we use the symbol hom to denote the morphism sets in Qcoh(X) as well as the induced functor to abelian groups. Further, the tensor product onQcoh(X) has a right adjoint functor denotedHomqc; see for example [23, 2.1].
We recall from [5, Definition 1.1, Proposition 1.3, and Definition 2.1]:
Definition 4.1. An acyclic complex F of flat-cotorsion sheaves on X is called totally acyclic if the complexes hom(C,F) and hom(F,C) are acyclic for every flat-cotorsion sheafC onX.
A sheafM onXis calledGorenstein flat-cotorsionif there exists a totally acyclic complexF of flat-cotorsion sheaves onX with M = Z0(F). Denote byGFC(X) the category of Gorenstein flat-cotorsion sheaves onX.1
We proceed to show that the sheaves defined in 4.1 are precisely the cotorsion Gorenstein flat sheaves, i.e. the sheaves that are both cotorsion and Gorenstein flat.
The next result is analogous to [5, Theorem 4.4].
Proposition 4.2. Assume that X is semi-separated noetherian. An acyclic com- plex of flat-cotorsion sheaves on X is totally acyclic if and only if it is F-totally acyclic.
Proof. LetF be a totally acyclic complex of flat-cotorsion sheaves. LetI be an injective sheaf and E an injective cogenerator in Qcoh(X). By [23, Lemma 3.2], the sheafHomqc(I,E) is flat-cotorsion. The adjunction isomorphism
hom(I ⊗F,E)∼= hom(F,Homqc(I,E)) (12)
along with faithful injectivity of E implies that I ⊗F is acyclic, hence F is F-totally acyclic.
For the converse, letF be anF-totally acyclic complex of flat-cotorsion sheaves andC be a flat-cotorsion sheaf. Recall from [23, Proposition 3.3] thatC is a direct summand ofHomqc(I,E) for some injective sheafI and injective cogeneratorE. Thus (12) shows that hom(F,C) is acyclic. Moreover, it follows from Theorem 3.3 that Zn(F) is cotorsion for everyn∈Z, so hom(C,F) is acyclic.
Theorem 4.3. Assume that X is semi-separated noetherian. A sheaf on X is Gorenstein flat-cotorsion if and only if it is cotorsion and Gorenstein flat; that is,
GFC(X) =Cot(X)∩GFlat(X).
Proof. The containment “⊆” is immediate by Theorem 3.3 and Proposition 4.2.
For the reverse containment, let M be a cotorsion Gorenstein flat sheaf on X. There exists an F-totally acyclic complex F of flat sheaves withM = Z0(F). As (CZac(Flat(X)),Csemi(Cot(X))) is a complete cotorsion pair, see Remark 3.1, there is an exact sequence inC(Qcoh(X))
0−→F −→T −→P−→0
withT ∈Csemi(Cot(X)) andP ∈CZac(Flat(X)). AsF andPareF-totally acyclic so isT; in particular, Zn(T) is cotorsion for everyn∈Z; see Theorem 3.3. The argument in [5, Theorem 5.2] now applies verbatim to finish the proof.
Remark 4.4. One upshot of Theorem 4.3 is that the Frobenius category described in Corollary 2.6 coincides with the one associated toGFC(X) per [5, Theorem 2.11].
In particular, the associated stable categories are equal. One of these is equivalent to the homotopy category of the Gorenstein flat model structure and the other is by [5, Corollary 3.9] and Proposition 4.2 equivalent to the homotopy category
KF-tac(Flat(X)∩Cot(X)) of F-totally acyclic complexes of flat-cotorsion sheaves onX.
1In [5] this category is denotedGorFlatCot(A) in the case of an affine schemeX= SpecA.
In [23, 2.5] thepure derived category of flat sheavesonX is the Verdier quotient D(Flat(X)) = K(Flat(X))
Kpac(Flat(X)),
whereKpac(Flat(X)) is the full subcategory ofK(Flat(X)) of pure acyclic complexes;
that is, the objects inKpac(Flat(X)) are precisely the objects inCZac(Flat(X)). Still following [23] we denote by DF-tac(Flat(X)) the full subcategory of D(Flat(X)) whose objects are F-totally acyclic. As the category Kpac(Flat(X)) is contained inKF-tac(Flat(X)), it can be expressed as the Verdier quotient
DF-tac(Flat(X)) = KF-tac(Flat(X)) Kpac(Flat(X)) .
Theorem 4.5. Assume that X is semi-separated noetherian. The composite of canonical functors
KF-tac(Flat(X)∩Cot(X))−→KF-tac(Flat(X))−→DF-tac(Flat(X)) is a triangulated equivalence of categories.
Proof. In view of Theorem 3.3 and the fact that (CZac(Flat(X)),Csemi(Cot(X))) is a complete cotorsion pair, see Remark 3.1, the proof of [5, Theorem 5.6] applies
mutatis mutandis.
We denote byStGFC(X) the stable category of Gorenstein flat-cotorsion sheaves;
cf. Remark 4.4. LetAbe a commutative noetherian ring of finite Krull dimension.
For the affine schemeX= SpecAthis category is by [5, Corollary 5.9] equivalent to the stable categoryStGPrj(A) of Gorenstein projectiveA-modules. This, together with the next result, suggests that StGFC(X) is a natural non-affine analogue of StGPrj(A). Indeed, the category DF-tac(Flat(X)) is Murfet and Salarian’s non- affine analogue of the homotopy category of totally acyclic complexes of projective modules; see [23, Lemma 4.22].
Corollary 4.6. There is a triangulated equivalence of categories StGFC(X)'DF-tac(Flat(X)).
Proof. Combine the equivalence StGFC(X) 'KF-tac(Flat(X)∩Cot(X)) from Re-
mark 4.4 with Theorem 4.5.
We emphasize that Proposition 4.2 and Theorem 4.5 offer another equivalent of the categoryStGFC(X), namely the homotopy category of totally acyclic complexes of flat-cotorsion sheaves.
A noetherian schemeX is called Gorenstein if the local ring OX,xis Gorenstein for everyx∈X. We close this paper with a characterization of Gorenstein schemes in terms of flat-cotorsion sheaves, it sharpens [23, Theorem 4.27]. In a paper in progress [3] we show that Gorensteinness of a schemeX can be characterized by the equivalence of the categoryStGFC(X) to a naturally defined singularity category.
Theorem 4.7. Assume thatX is semi-separated noetherian. The following condi- tions are equivalent.
(i) X is Gorenstein.
(ii) Every acyclic complex of flat sheaves onX is F-totally acyclic.
(iii) Every acyclic complex of flat-cotorsion sheaves on X is F-totally acyclic.
(iv) Every acyclic complex of flat-cotorsion sheaves on X is totally acyclic.
Proof. The equivalence of conditions (i) and (ii) is [23, Theorem 4.27], and con- ditions (iii) and (iv) are equivalent by Proposition 4.2. As (ii) evidently implies (iii), it suffices to argue the converse.
Assume that every acyclic complex of flat-cotorsion sheaves isF-totally acyclic.
LetF be an acyclic complex of flat sheaves. As (CZac(Flat(X)),Csemi(Cot(X))) is a complete cotorsion pair, see Remark 3.1, there is an exact sequence inC(Qcoh(X)),
0−→F −→C −→P −→0,
with C ∈ Csemi(Cot(X)) and P ∈ CZac(Flat(X)). Since F and P are acyclic, the complexC is also acyclic. Moreover,C is a complex of flat-cotorsion sheaves.
By assumption C is F-totally acyclic, and so is P, whence it follows that F is
F-totally acyclic.
Acknowledgment
We thank Alexander Sl´avik for helping us correct a mistake in an earlier version of the proof of Theorem 1.6. We also acknowledge the anonymous referee’s prompts to improve the presentation.
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L.W.C. Texas Tech University, Lubbock, TX 79409, U.S.A.
Email address:[email protected] URL:http://www.math.ttu.edu/~lchriste
S.E. Universidad de Murcia, Murcia 30100, Spain Email address:[email protected]
URL:https://webs.um.es/sestrada/
P.T. Norwegian University of Science and Technology, 7491 Trondheim, Norway Email address:[email protected]
URL:https://folk.ntnu.no/pedertho
